Systems and Methods for Quantum Tomography Using an Ancilla

ABSTRACT

Quantum computing systems and methods are provided. In one example, a quantum computing system includes a quantum system having one or more quantum system qubits and one or more ancilla qubits. The quantum computing system includes one or more quantum gates implemented by the quantum computing system. The quantum gate(s) are operable to configure the one or more ancilla qubits into a known state. The quantum computing system includes a quantum measurement circuit operable to perform a plurality of measurements on the one or more quantum system qubits using the one or more ancilla qubits. The quantum computing system includes one or more processors operable to determine a reduced density matrix for a subset of the quantum system based on a set of the plurality of measurements that include a number of repeated measurements performed using the quantum measurement circuit.

PRIORITY CLAIM

The present application claims the benefit of priority of U.S.Provisional Application Ser. No. 62/915,246, titled “Systems and Methodsfor Quantum Tomography Using an Ancilla,” filed on Oct. 15, 2019, whichis incorporated herein by reference.

FIELD

The present disclosure relates generally to quantum computing systems.

BACKGROUND

Quantum computing is a computing method that takes advantage of quantumeffects, such as superposition of basis states and entanglement toperform certain computations more efficiently than a classical digitalcomputer. In contrast to a digital computer, which stores andmanipulates information in the form of bits, e.g., a “1” or “0,” quantumcomputing systems can manipulate information using quantum bits(“qubits”). A qubit can refer to a quantum device that enables thesuperposition of multiple states, e.g., data in both the “0” and “1”state, and/or to the superposition of data, itself, in the multiplestates. In accordance with conventional terminology, the superpositionof a “0” and “1” state in a quantum system may be represented, e.g., asa |0>+b |1> The “0” and “1” states of a digital computer are analogousto the |0> and |1> basis states, respectively of a qubit.

SUMMARY

Aspects and advantages of embodiments of the present disclosure will beset forth in part in the following description, or can be learned fromthe description, or can be learned through practice of the embodiments.

One example aspect of the present disclosure is directed to a quantumcomputing system. The quantum computing system includes a quantum systemhaving one or more quantum system qubits and one or more ancilla qubits.Each quantum system qubit is associated with one of the one or moreancilla qubits. The quantum computing system includes one or morequantum gates implemented by the quantum computing system. The one ormore quantum gates are operable to configure the one or more ancillaqubits into a known state. The quantum computing system includes aquantum measurement circuit implemented by the quantum computing system.The quantum measurement circuit is operable to perform a plurality ofmeasurements on the one or more quantum system qubits using the ancillaqubits. The quantum computing system includes one or more processorsoperable to perform operations. The operations include determining areduced density matrix for a subset of the quantum system based on a setof the plurality of measurements. The set of the plurality ofmeasurements include a number of repeated measurements performed on thequantum system qubits using the quantum measurement circuit. Theoperations include outputting data indicative of the reduced densitymatrix.

Other aspects of the present disclosure are directed to various systems,methods, apparatuses, non-transitory computer-readable media,computer-readable instructions, and computing devices.

These and other features, aspects, and advantages of various embodimentsof the present disclosure will become better understood with referenceto the following description and appended claims. The accompanyingdrawings, which are incorporated in and constitute a part of thisspecification, illustrate example embodiments of the present disclosureand, together with the description, serve to explain the relatedprinciples.

BRIEF DESCRIPTION OF THE DRAWINGS

Detailed discussion of embodiments directed to one of ordinary skill inthe art is set forth in the specification, which makes reference to theappended figures, in which:

FIG. 1 depicts an example quantum computing system according to exampleembodiments of the present disclosure;

FIG. 2 depicts an overview of determining reduced density matrices for aquantum system according to example embodiments of the presentdisclosure;

FIG. 3 depicts quantum gates and a quantum measurement circuit accordingto example embodiments of the present disclosure;

FIG. 4 an example fermion-to-qubit mapping according to exampleembodiments of the present disclosure;

FIG. 5 depicts a flow diagram of an example method according to exampleembodiments of the present disclosure;

FIG. 6 depicts a flow diagram of an example method according to exampleembodiments of the present disclosure; and

FIG. 7 depicts a flow diagram of an example method according to exampleembodiments of the present disclosure.

DETAILED DESCRIPTION

Example aspects of the present disclosure are directed to quantumcomputing systems and methods that are operable to determine aspects ofa state of a quantum system by performing repeated measurements of thequantum system using a quantum measurement circuit (e.g., a singlequantum measurement circuit). In some implementations, reduced densitymatrix(s) for a quantum system can be determined by performing repeatedmeasurements using the quantum measurement circuit of one or more qubitsin the quantum system a specific number of times. In particular aspects,the specific number of repeated measurements can be determinedindependent of a size (e.g., number of qubits) of the quantum system.The specific number of repeated measurements can scale as a function ofdesired precision of the reduced density matrix(s). The systems andmethods can be applied to the general case of determining aspects of astate of a quantum system having a plurality of qudits. The system andmethods can be used to determine elements of fermionic reduced densitymatrices in fermionic quantum systems.

Reduced density matrices can be used in quantum tomography applicationsto describe a state of a subset of a quantum system. For instance, ak-qubit reduced density matrix can describe a correlation of k qubits ina quantum system having n total qubits, wherein k is less than n.Determining reduced density matrices from measurement data of a quantumsystem can be a relevant task when assessing a state of a quantum systemusing quantum computing systems. For instance, reduced density matricescan provide information regarding correlations between a subset ofqubits and/or particles in a quantum system that can allow forcomputation and/or analysis of various properties of the quantum system,such as monopole moments and derivatives of energy.

According to example aspects of the present disclosure, all elements forall k-qubit reduced density matrices for a quantum system can bedetermined to a desired precision from a number of repeated measurementsperformed using a quantum measurement circuit (e.g., a single quantummeasurement circuit) on one or more quantum qubits in the quantumsystem. In particular implementations, the number of repeatedmeasurements can be about 3^(k)/∈², where k is indicative of a number ofqubits in a subset of qubits in the quantum system represented by thek-qubit reduced density matrix and c is the desired precision (e.g.,expresses as a standard deviation). Thus, the number of repeatedmeasurements can be determined irrespective of a total number n ofqubits in the quantum system.

More particularly, a quantum system can include one or more quantumsystem qubits. Aspects of the present disclosure can attach or associatean ancilla qubit with each of the one or more quantum system qubits.Quantum gate(s) implemented in the quantum computing system canconfigure the ancilla qubits into a known state. A quantum measurementcircuit implemented in the quantum computing system can perform aplurality of measurements (e.g., Bell-basis measurements). Themeasurements can be performed in parallel for each quantum system qubitin the quantum system. For example, the system may comprise a pluralityof system qubit-ancilla qubit pairs, and measurements for each systemqubit can be performed by analyzing the state of the relevant pair (forexample, measuring the pair in the Bell basis). A reduced density matrixassociated with a subset of the quantum system (e.g., a k-qubit reduceddensity matrix for the quantum system having n qubits where k<n) can bedetermined based on a set of the plurality of measurements. In someembodiments, all k-qubit reduced density matrices for the quantum system(e.g., k-qubit reduced density matrices for all different subsets ofk-qubits in the quantum system) can be determined based on the set ofthe plurality of measurements.

The set of the plurality of measurements can include a number ofrepeated measurements performed on the quantum system qubits using thequantum measurement circuit. The number of repeated measurements can bedetermined as a function of the number of qubits (e.g., k) in the subsetof the quantum system to be described by the reduced density matrices.In this way, quantum tomography techniques to determine a state of aquantum system using reduced density matrices can be implemented with asingle quantum measurement circuit and with a reduced number of repeatedmeasurements independent of quantum system size.

The quantum computing systems and methods according to example aspectsof the present disclosure are applicable to the general case of qudits.For instance, a quantum system can have one or more qudits where eachqudit has a plurality of quantum levels (e.g., two quantum levels in thecase of a qubit, three quantum levels, four quantum levels, and soforth). All elements for all k-qudit reduced density matrices for aquantum system can be determined to a desired precision from a number ofrepeated measurements performed using a quantum measurement circuit(e.g., a single quantum measurement circuit) on one or more quantumsystem qudits in the quantum system. In particular implementations, thenumber of repeated measurements can be about (D+1)^(k)/∈², where D isindicative of a number of quantum levels in the qudit(s), k isindicative of a number of qudits in a subset of qubits in the quantumsystem represented by the k-qudit reduced density matrix, and c is thedesired precision. The number of repeated measurements can be determinedirrespective of a total number of qudits in the quantum system.

The quantum computing systems and methods according to example aspectsof the present disclosure are applicable to determining k-particlefermionic reduced density matrices for a quantum system. Simulatingcorrelated fermionic systems can be a relevant application of quantumcomputing systems. For instance, estimating the energy of a quantumstate can require a partial quantum tomography of two particle fermionicreduced density matrices. Two particle fermionic reduced densitymatrices can also be used to derive a number of properties of materials.In addition, two-particle reduced density matrices can reduce basiserror by relaxing single particle orbitals.

The quantum computing systems and methods according to example aspectsof the present disclosure can implement a ternary tree mapping as afermion-to-qubit mapping. This fermion-to-qubit mapping can provide anoptimal and/or enhanced fermion-to-qubit mapping for many quantumcomputing applications. As one example, all elements for all k-particlefermionic reduced density matrices for a fermionic quantum system can bedetermined to a desired precision from a number of repeated measurementsperformed using a quantum measurement circuit (e.g., a single quantummeasurement circuit) on one or more quantum system qubits in the quantumsystem. In particular implementations, the number of repeatedmeasurements can be about (2n+1)^(k)/ε², where n is the total number ofquantum system qubits in the quantum system, k is indicative of a numberof fermionic modes in a subset of the quantum system represented by thek-particle reduced density matrix, and c is the desired precision. Thefermion-to-qubit mapping using a ternary tree mapping can be used forother quantum computing applications without deviating from the scope ofthe present disclosure.

Aspects of the present disclosure can provide a number of technicaleffects and benefits and can provide improvements to quantum computingtechnology. For instance, quantum tomography can be more efficientlyimplemented using a single quantum measurement circuit with a reducednumber of repeated measurements irrespective of quantum system size.This can lead to faster determination of reduced density matrices and/orthe ability to perform more measurements within a given period of timeto achieve increased precision. Moreover, a bottleneck of currentquantum computing devices is a low repetition rate which can constrainthe number of repeated measurements that can be accurately performed ina given time frame before significant errors are introduced into thesystem. Aspects of the present disclosure are more readily implementedwithin the constraints of current quantum devices by requiring fewerrepeated measurements using a single quantum measurement circuit. Inaddition, use of a ternary tree mapping as a quantum

With reference now to the FIGS., example embodiments of the presentdisclosure will be discussed in further detail. As used here, the use ofthe term “about” in conjunction with a value refers to within 20% of thevalue.

FIG. 1 depicts an example quantum computing system 100. The examplesystem 100 is an example of a system implemented as classical or quantumcomputer program on one or more classical computers or quantum computingdevices in one or more locations, in which the systems, components, andtechniques described below can be implemented. FIG. 1 depicts an examplequantum computing system that can be used to implement aspects of thepresent disclosure. Those of ordinary skill in the art, using thedisclosures provided herein, will understand that other quantumcomputing structures or system can be used without deviating from thescope of the present disclosure.

The system 100 includes quantum hardware 102 in data communication withone or more classical processors 104. The quantum hardware 102 includescomponents for performing quantum computation. For example, the quantumhardware 102 includes a quantum system 110, control device(s) 112, andreadout resonator(s) 114. The quantum system 110 can include one or moremulti-level quantum subsystems, such as a register of qubits. In someimplementations, the multi-level quantum subsystems can includesuperconducting qubits, such as flux qubits, charge qubits, transmonqubits, etc. In some implementations, the multi-level quantum subsystemscan include one or more qudits (e.g., units of quantum informationdescribed by superposition of D states). In some implementations, themulti-level quantum subsystems can include fermionic quantum subsystems.

The type of multi-level quantum subsystems that the system 100 utilizesmay vary. For example, in some cases it may be convenient to include oneor more readout resonators 114 attached to one or more superconductingqubits, e.g., transmon, flux, Gmon, Xmon, or other qubits. In othercases ion traps, photonic devices or superconducting cavities (withwhich states may be prepared without requiring qubits) may be used.Further examples of realizations of multi-level quantum subsystemsinclude fluxmon qubits, silicon quantum dots or phosphorus impurityqubits.

Quantum circuits may be constructed and applied to the register ofqubits included in the quantum system 110 via multiple control linesthat are coupled to one or more control devices 112. Example controldevices 112 that operate on the register of qubits include quantum logicgates or circuits of quantum logic gates, e.g., Hadamard gates,controlled-NOT (CNOT) gates, controlled-phase gates, or T gates. In someimplementations, T gates may be stored in one or more T factoriesincluded in the quantum hardware 102. The one or more control devices112 may be configured to operate on the quantum system 110 through oneor more respective control parameters (e.g., one or more physicalcontrol parameters). For example, in some implementations, themulti-level quantum subsystems may be superconducting qubits and thecontrol devices 112 may include one or more digital to analog converters(DACs) with respective voltage physical control parameters.

The quantum hardware 102 may further include measurement devices, e.g.,readout resonators 114. Measurement results 108 obtained via measurementdevices may be provided to the classical processors 104 for processingand analyzing. In some implementations, the quantum hardware 102 mayinclude a quantum circuit and the control device(s) 112 and readoutresonator(s) 114 may include one or more quantum logic gates thatoperate on the quantum system 102 through microwave pulse physicalcontrol parameters that are sent through wires included in the quantumhardware 102. Further examples of control devices include arbitrarywaveform generators, wherein a DAC creates the signal. The controlparameters may include qubit frequencies.

The readout resonator(s) 114 may be configured to perform quantummeasurements on the quantum system 110 and send measurement results 108to the classical processors 104. In addition, the quantum hardware 102may be configured to receive data specifying physical control parametervalues 106 from the classical processors 104. The quantum hardware 102may use the received physical control parameter values 106 to update theaction of the control device(s) 112 and readout resonator(s) 114 on thequantum system 110. For example, the quantum hardware 102 may receivedata specifying new values representing voltage strengths of one or moreDACs included in the control devices 112 and may update the action ofthe DACs on the quantum system 110 accordingly.

The classical processors 104 may be configured to initialize the quantumsystem 110 in an initial quantum state, e.g., by sending data to thequantum hardware 102 specifying an initial set of parameters 106.

The readout resonator 114 can take advantage of a difference in theimpedance for the |0> and |1> states of an element of the quantumsystem, such as a qubit, to measure the state of the element (e.g., thequbit). For example, the resonance frequency of the readout resonator114 can take on different values when a qubit is in the state |0> or thestate |1>, due to the nonlinearity of the qubit. Therefore, a microwavepulse reflected from the readout resonator 114 carries an amplitude andphase shift that depend on the qubit state. In some implementations, aPurcell filter can be used in conjunction with the readout resonator 114to impede microwave propagation at the qubit frequency.

FIG. 2 depicts an overview of determining aspects of a quantum systemusing reduced density matrices according example aspects of the presentdisclosure. Given an n-qubit quantum state p the k-qubit reduced densitymatrix (k-RDM) may be written as:

$\begin{matrix}{{{Tr}_{{\neq j_{1}},\ldots,j_{k}}(\rho)} = {\frac{1}{2^{k}}{\sum_{\alpha_{1},\ldots,{\alpha_{k} = 0},x,y,z}{\rho_{j_{1},\ldots,j_{k}}^{\alpha_{1},\ldots,\alpha_{k}} \otimes_{i = 1}^{k}\sigma_{j_{i}}^{\alpha_{i}}}}}} & (1)\end{matrix}$

where σ_(j) ^(a) is the Pauli operator on the j-th qubit, σ⁰=andσ^(x,y,z) are the Pauli-x, -y, and -z operators, respectively. Thek-qubit correlation functions in Eq. (1), also referred to as k-RDMs,are defined as

$\begin{matrix}{\rho_{j_{1},\ldots,j_{k}}^{\alpha_{1},\ldots,\alpha_{k}} = {{{Tr}\left( {\rho \otimes_{i = 1}^{k}\sigma_{j_{i}}^{\alpha_{i}}} \right)}.}} & (2)\end{matrix}$

Measuring the correlation functions for all α₁, . . . , α_(k) and allj₁, . . . , j_(k) amounts to determining all k-RDMs. Assuming all(k−1)-RDMs are known, measuring the (n_(k)) 3^(k) different observables,⊗_(i=1) ^(k)σ_(ji) ^(αj)(α_(i)≠0), provides the required information toreconstruct all k-RDMs. Under the assumption that only single-qubitoperations are allowed, it has been shown that elements in all k-RDMscan be sampled by implementing e^(O(k))log(n) different quantumcircuits. By repeating each circuit 1/∈² times, the statistical error inestimating these elements scale as ∈, where ∈ is precision.

Changing quantum circuits is typically quite slow on certainprogrammable quantum devices (e.g., those based on FPGAs), whilerepeating a single circuit may be done much faster. To circumvent theproblem of programming various circuits, the quantum computing systemsand methods according to example aspects of the present disclosure canallow estimation of elements in all k-qubit RDMs using a single quantumcircuit, with error that scales as E by running it for about 3^(k)/∈²times.

More particularly, as shown in FIG. 2 , a state of a quantum system 302(e.g., one or more qubits in quantum system 110 of FIG. 1 ) can bedetermined by performing a plurality of repeated measurements 304 of thequantum system 302 with a quantum measurement circuit. Moreparticularly, an ancilla qubit j′ can be associated with each of the oneor more quantum system qubits j in the quantum system 302. The ancillaqubit j′ can be configured in a known state μ (e.g., using one or morequantum gates), so that the total system-ancilla state is ρ ⊗ μ^(⊗n).Each pair of qubits j, j′ can be measured in the common eigenbasis ofσ^(x) ⊗ σ^(x), σ^(y) ⊗ σ^(y), and σ^(z) ⊗ σ^(z), i.e., the Bell basis.The plurality of measurements can be repeated for each pair of qubits j,j′ in the quantum system 302 in parallel.

FIG. 3 depicts a wire representation of an example quantum circuit 320that can be used to perform the plurality of repeated measurements 304.The wire 322 is representative of the ancilla qubit. The wire 325 isrepresentative of the quantum system qubit. As shown, quantum gate(s)330 can be used to configure the ancilla qubit in a known state μ byrotating |0> about an x axis by an angle θ and a rotation around the zaxis by an angle ϕ. Rotation about the x axis can be implemented byquantum gate 332. Rotation about the z axis can be implemented byquantum gate 334. If the Pauli-x, -y, and -z operators are sampled atthe same rate, a natural choice of the ancilla state is

$\begin{matrix}{{\mu = {\frac{1}{2}\left( {{+ \frac{1}{\sqrt{3}}}\left( {\sigma^{x} + \sigma^{y} + \sigma^{z}} \right)} \right)}},} & (3)\end{matrix}$

which leads to Tr(σ^(x)μ)=Tr(σ^(y)μ)=Tr(σ^(z)μ)=1/√{square root over(3)}. In that regard, in some embodiments, θ is equal to about arccos1/√{square root over (3)} and ϕ is equal to about 3π/4.

The quantum measurement circuit 340 of FIG. 3 can be configured to takeBell-basis measurements of each qubit pair j,j′ using Hadamard gate 342and CNOT gate 344 to provide Bell-basis measurements 346 and 348.Measuring the system and ancilla qubits in the Bell basis yields

$\begin{matrix}{{{Tr}\left( {{\rho \otimes \mu^{\otimes n}}{\prod_{i = 1}^{k}{\sigma_{j_{i}}^{\alpha_{i}} \otimes \sigma_{j_{i}^{\prime}}^{\alpha_{i}}}}} \right)} = {\frac{1}{{\sqrt{3}}^{k}}{\rho_{j_{1},\ldots,j_{k}}^{\alpha_{1},\ldots,a_{k}}.}}} & (4)\end{matrix}$

Due to the factor 1/√{square root over (3)}^(k), we must run theexperiment about 3^(k)/∈² times to obtain the standard-deviation error∈.

Accordingly, referring back to FIG. 2 , all k-qubit reduced densitymatrices 308 can be determined by processing a set of repeatedmeasurements 306 performed as described with reference to FIG. 3 . Theset of repeated measurements can be about 3^(k)/∈², where k isindicative of a number of qubits in the subset of the quantum systemrepresented by the k-qubit reduced density matrix and ∈ is a desiredprecision of the k-qubit reduced density matrix.

In some embodiments, the factors k and ∈ can be obtained (e.g., providedas an input) by the quantum computing system (e.g., as specifiedparameters). The quantum computing system can automatically determinethe number of repeated measurements to be performed on the quantumsystem as shown at 310 and the quantum computing system can becontrolled to perform the number of repeated measurements 304 on thequantum system 302 to determine all k-qubit reduced density matrices forthe quantum system.

The scheme for providing a state of a quantum system having one or morequbits described with reference to FIGS. 3 and 4 can be generalized ton-qudit systems. Each qudit can have a D quantum levels (e.g., twolevels in the case of a qubit, three quantum levels, etc.). Oneapplication is to collection of D-level spin systems.

Consider the Heisenberg-Weyl (HW) shift and phase (clock) operators,

$\begin{matrix}\begin{matrix}{\left. {\left. {X{❘d}} \right\rangle = {❘{d \oplus 1}}} \right\rangle,} & {\left. {\left. {Z{❘d}} \right\rangle = {e^{\frac{2\pi{id}}{D}}{❘d}}} \right\rangle,}\end{matrix} & (5)\end{matrix}$ whered = 0, …, D − 1

and ⊗ is addition modulo D. The operators obey the HW commutationrelation

${XZ} = {e^{\frac{2\pi i}{D}}{{ZX}.}}$

Similarly to the Pauli operators on a qubit, the HW operators generate acomplete operator basis acting on the D-dimensional Hilbert space. Inparticular, a density matrix p of a qubit can be expressed as

$\begin{matrix}{{\rho = {\frac{1}{D}\left( {+ {\sum_{f \land {g \neq 0}}^{D - 1}\begin{matrix}\rho^{f} & {{\,^{g}X^{f}}Z^{g}}\end{matrix}}} \right)}},} & (6)\end{matrix}$ ${where}\begin{matrix}\rho^{f} & {\,^{g}{= {{Tr}\left( {Z^{- g}X^{- f}\rho} \right)}}}\end{matrix}$

are the expansion coefficients. Similarly, given an n-qudit densitymatrix ρ, a k-qudit RDM can be written as

$\begin{matrix}{{{T{r_{{\neq j_{1}},\ldots,j_{k}}(\rho)}} = {\frac{1}{D^{k}}{\sum_{f_{1},{g_{1}\ldots},f_{k},{g_{k} = 0}}^{D - 1}{{\rho_{j_{1},\ldots,j_{k}}^{{f_{1}g_{1}},\ldots,{f_{k}g_{k}}} \otimes_{i = 1}^{k}X_{j_{i}}^{f_{i}}}Z_{j_{i}}^{g_{i}}}}}},} & (7)\end{matrix}$

where X_(ji) ^(f) ^(i) Z_(ji) ^(gi) are the HW operators on the j_(i)-thqudit, and

$\begin{matrix}{{\rho_{j_{1},\ldots,j_{k}}^{{f_{1}g_{1}},\ldots,{f_{k}g_{k}}} = {{Tr}\left( {{\rho \otimes_{i = 1}^{k}X_{j_{i}}^{f_{i}}}Z_{j_{i}}^{g_{i}}} \right)}},} & (8)\end{matrix}$

By measuring the above correlation functions for all f₁, h₁ . . . f_(k),h_(k) and all j₁, . . . j_(k) we can determine all k-qudit RDMs.

While the HW operators on a qudit are not commute or anticommute as thePauli operators on a qubit, nevertheless, all the required correlationfunctions can be measured with a single quantum circuit according toexample aspects of the present disclosure. Consider the generalizationof the Bell-basis to qudits,

$\begin{matrix}{\left. {\left. {\left. {❘\Phi_{h\ell}} \right\rangle = {\otimes {X^{h}Z^{\ell}{❘\Phi_{00}}}}} \right\rangle = {Z^{\ell}{X^{- h} \otimes {❘\Phi_{00}}}}} \right\rangle,} & (9)\end{matrix}$ where $\begin{matrix}{\left. {\left. {\left. {❘\Phi_{00}} \right\rangle = {\frac{1}{\sqrt{D}}{\sum_{d = 0}^{D - 1}{❘d}}}} \right\rangle \otimes {❘d}} \right\rangle.} & (10)\end{matrix}$

Following the HW commutation relation we obtain,

=

ϕ₀₀|1⊗(X^(h)

)+X^(h′)

|ϕ₀₀>  (11)

=

δ_(h,h′),   (12)

where the

=1

and

=0 otherwise (and similarly for δ_(h,h′)). Therefore, the set {|

>: h,

=0, . . . , D−1} form an orthonormal basis for two qudits. It is theeigenbasis of the operators X^(f)Z^(g) ⊗X^(f)Z^(−g) for f, g=0, . . . ,D−1,

X^(f)Z^(g) ⊗X^(f)Z^(−g)|

>=

|

>.   (13)

Hence, measuring these mutually-commuting operators in the generalizedBell basis (e.g., equation (9)) reveals their values simultaneously.

Similar to the qubit case, one can estimate elements in all k-qudit RDMswith error ∈ by repeating the following steps for (D+1)^(k)/∈² times:(1) to each system qudit (labeled by j) attach an ancillary qudit(labelled by j′) in a known state μ, so that the total system-ancillastate is ρ⊗μ^(⊗n); (2) measure each pair of qudits (j,j′) in the commoneigenbasis of X^(f)Z^(g) ⊗X^(f) Z^(−g), i.e., the generalized Bellbasis.

By choosing a qudit ancilla state μ such that

$\begin{matrix}{{{T{r\left( {X_{j_{i}^{\prime}}^{f_{i}}Z_{j_{i}^{\prime}}^{- g_{i}}\mu} \right)}} = \frac{1}{\sqrt{D + 1}}},} & (14)\end{matrix}$

the following is obtained:

$\begin{matrix}{\rho_{j_{1},\ldots,j_{k}}^{{f_{1}g_{1}},\ldots,{f_{k}g_{k}}} = {\frac{{Tr}\left( {{\rho \otimes \mu^{\otimes n}}{\prod_{i = 1}^{k}{X_{j_{i}}^{f_{i}}{Z_{j_{i}}^{gi} \otimes X_{j_{i}^{\prime}}^{f_{i}}}Z_{j_{i}^{\prime}}^{- {gi}}}}} \right)}{{\sqrt{D + 1}}^{k}}.}} & (15)\end{matrix}$

The last equation implies that to obtain a fixed standard-deviationerror ∈ in the measured quantities, the number of repeated measurementsis about (D+1)^(k)/∈² times.

The quantum computing systems and methods described above can also beapplied to fermionic quantum systems. One way to estimate the energy ofa quantum state in a variational quantum algorithm is by measuring a setof observables that include the fermionic k-particle reduced densitymatrix. The fermionic k-particle reduced density matrix can include

(n^(2k)) parameters, from which a large number of properties of aquantum system can be determined. Determining elements in the reduceddensity matrices also allows one to implement error mitigation schemesbased on subspace expansions.

In the second quantization formalism, the elements in a k-particlereduced density matrix of an n-mode fermionic system are expectationvalues involving 2k fermionic operators. For instance, the fermionic2-reduced density matrix may be written as

ρ_(pqrs)=

Y_(p)Y_(q)Y_(r)Y_(s)

,   (16)

where the γ's are the Majorana fermion operators

$\begin{matrix}{\begin{matrix}{{\gamma_{2j} = {\frac{1}{\sqrt{2}}\left( {c_{j}^{\dagger} + c_{j}} \right)}},} & {\gamma_{{2j} + 1} = {\frac{i}{\sqrt{2}}\left( {c_{j}^{\dagger} - {ic_{j}}} \right)}}\end{matrix},} & (17)\end{matrix}$

for j=1,2, . . . , n. The 2n Majorana operators mutually anticommute,i.e., {y_(p), y_(q)}=0 for p≠q.

There are various ways to map fermionic operators to qubit operators.The Jordan-Wigner transformation (JWT) maps single fermionic operatorsto qubit operators acting on

(n) qubits. In comparison, the Bravyi-Kitaev transformation (BKT)reduces that number to

(log n) qubits. The three Pauli operators are neither balanced in theJWT or BKT, as the Pauli-z operators are used more frequently in theJWT, while the Pauli-x and -z operators are used more often in the BKT.

Example aspects of the present disclosure implement a fermion-to-qubitmapping that maps single fermionic operators to qubit operators withbalanced numbers of Pauli-x, -y, and -z operators. For instance, thefermion-to-qubit mapping can be defined on a ternary tree, where a qubitis put on each node of the tree that is not a leaf. The tree can becomplete, i.e., all levels can be fully filled. The total number ofqubits, i.e., nodes excluding the leaves, in a ternary tree of height his

$\begin{matrix}{n = {{\sum_{\ell = 0}^{h - 1}3^{\ell}} = {\frac{3^{h} - 1}{2}.}}} & (18)\end{matrix}$

FIG. 4 depicts a ternary tree mapping used to implement afermion-to-qubit mapping 350 according to example embodiments of thepresent disclosure. The qubit associated with the root node 352 islabeled 0. The rest of the qubits are indexed consecutively going downthe tree as shown in FIG. 4 . The Pauli operators of the η-th qubit aredenoted as σ_(η) ^(x,y,z). Each root-to-leaf path on the tree can beuniquely specified by the vector p=(p₀, . . . , ρ_(h−1)), where

=0, 1, 2 determines the next node on the path with depth

+1 based on the current node with depth

. The index of a node of depth

on the path p can be written by

η(p,

)=(

−1)/2+

p_(j), for

>0,   (19)

where (

−1)/2 is the number of nodes with depth less than

. An operator for each root-to-leaf path p can be introduced:

, A_(p) ²=,   (20)

where

(p)=x, y, z for p=0, 1, 2, respectively. By construction these operatorsare mutually anticommute, i.e., {A_(p), A_(q)}=for p≠q. This is becausep and q have to diverge at some point. Before that point they involvethe same Pauli operators on the same qubits, at the point of divergenceA_(p) and A_(q) associate different Pauli operator to the same qubit,whereas after that point they involve Pauli operators on differentqubits.

There are 3^(h)=2n+1 distinct root-to-leaf paths in the ternary tree,whereas the total number of independent operators A_(p) is 2n. This isbecause the product of A_(p) for all paths p is proportional to theidentity operator. Therefore, 2n Majorana operators can be mapped to 2nindependent Pauli operators A_(p) with the same anticommutationrelations. The Pauli weight of A_(p) equals to the tree heighth=log₃(2n+1), which is lower than log₂(n+1) in the BKT. As a result, thePauli weights of elements in fermionic k-RDMs are at most 2 klog₃(2n+1). Using quantum measurement according to example aspects of thepresent disclosure, the matrix elements can be attenuated by a factorbounded by

√{square root over (3)}^(2klog) ³ ^((2n+1))=(2n+1)^(k).   (21)

Therefore, all fermionic k-RDMs can be measured to precision ∈ byrepeating the same circuit according to example aspects of the presentdisclosure for about (2n+1)^(k)/∈² times.

FIG. 5 depicts a flow diagram of an example method 500 according toexample embodiments of the present disclosure. The method 500 can beimplemented using any suitable quantum computing system, such as thequantum computing system 100 depicted in FIG. 1 . FIG. 5 depicts stepsperformed in a particular order for purposes of illustration anddiscussion. Those of ordinary skill in the art, using the disclosuresprovided herein, will understand that various steps of any of themethods disclosed herein can be adapted, modified, performedsimultaneously, omitted, include steps not illustrated, rearranged,and/or expanded in various ways without deviating from the scope of thepresent disclosure.

At 502, the method can include accessing a quantum system (e.g., thequantum system 102 of FIG. 1 ). The quantum system can include one ormore quantum system qubits. The quantum system can include one or moreancilla qubits. Each ancilla qubit can be associated with one of the oneor more quantum system qubits.

At 504, the method can including configuring each ancilla qubit in aknown state. For instance, as discussed with reference to FIGS. 2 and 3, quantum gate(s) implemented by a quantum computing system canconfigure each ancilla qubit in the quantum system in a known state. Forthe case of qubits, each ancilla qubit can be configured in a knownstate μ (e.g., using one or more quantum gates), so that the totalsystem-ancilla state is ρ⊗μ^(⊗n).

In some embodiments, the ancilla qubit can be configured in a knownstate μ by rotating |0> about an x axis by an angle θ followed by arotation around the z axis by an angle ϕ. If the Pauli-x, -y, and -zoperators are sampled at the same rate, a natural choice of the ancillastate is

$\begin{matrix}{{\mu = {\frac{1}{2}\left( {{+ \frac{1}{\sqrt{3}}}\left( {\sigma^{x} + \sigma^{y} + \sigma^{z}} \right)} \right)}},} & (3)\end{matrix}$

which leads to Tr(σ^(x)μ)=Tr(σ^(y)μ)=Tr(σ^(z)μ)=1/√{square root over(3)}. In that regard, in some embodiments, θ is equal to about arccos1/√{square root over (3)} and ϕ is equal to about 3π/4.

At 506 of FIG. 5 , the method can include obtaining a plurality ofmeasurements (e.g., Bell-basis measurements) using a quantum measurementcircuit. The plurality of measurements can be performed, for instance,using the quantum circuit 320 depicted in FIG. 3 . The plurality ofmeasurements can be repeated a specific number of times. Measurementsfor each qubit and ancilla qubit pair in the quantum system can beperformed in parallel with the quantum measurement circuit.

At 508, the method can include determining reduced density matrix(s)from a set of the plurality of measurements. For instance, the classicalprocessor(s) 104 of FIG. 1 can determine reduce density matrix(s) from aset of the plurality of measurements. The set of plurality ofmeasurements can include a number of repeated measurements performed onthe quantum system In some embodiments, the reduced density matrix(s)can include all k-qubit reduced density matrices for the quantum system.The number of repeated measurements can be determined as a function of asize of the subset of the quantum system described by the reduceddensity matrix irrespective of a number of quantum system qubits in thequantum system. The number of repeated measurements can be about3^(k)/∈², where k is indicative of a number of qubits in the subset ofthe quantum system represented by the k-qubit reduced density matrix and∈ is a desired precision of the k-qubit reduced density matrix.

At 510, the method can include outputting data indicative of the reduceddensity matrix(s). For example, the data indicative of the reduceddensity matrix(s) can be communicated to another component of thequantum computing system (e.g., another processor) or to a remotedevice. As another example, data indicative of the reduced densitymatrix(s) can be stored in one or more memory devices. As anotherexample, data indicative of the reduced density matrix(s) can beprovided for display on one or more display devices. As another example,data indicative of the reduced density matrix(s) can be provided foraccess by one or more computer programs, modules, executableinstructions, or the like.

FIG. 6 depicts a flow diagram of an example method 600 according toexample embodiments of the present disclosure. The method 600 can beimplemented using any suitable quantum computing system, such as thequantum computing system 100 depicted in FIG. 1 . FIG. 6 depicts stepsperformed in a particular order for purposes of illustration anddiscussion. Those of ordinary skill in the art, using the disclosuresprovided herein, will understand that various steps of any of themethods disclosed herein can be adapted, modified, performedsimultaneously, omitted, include steps not illustrated, rearranged,and/or expanded in various ways without deviating from the scope of thepresent disclosure.

At 602, the method can include accessing a quantum system (e.g., thequantum system 102 of FIG. 1 ). The quantum system can include one ormore quantum system qudits. The quantum system can include one or moreancilla qudits. Each ancilla qudit can be associated with one of the oneor more quantum system qudits. Each qudit can be associated with aplurality of quantum levels. For instance, each qudit can be associatedwith D quantum levels.

At 604, the method can including configuring each ancilla qudit in aknown state. For instance, quantum gate(s) implemented by a quantumcomputing system can configure each ancilla qudit in the quantum systemin a known state. For the case of qudits, each ancilla qudit can beconfigured in a known state μ as described with reference to equations(13) and (14) above.

At 606, the method can include obtaining a plurality of measurements(e.g., Bell-basis measurements) using a quantum measurement circuit. Theplurality of measurements can be repeated a specific number of times.Measurements for each qudit and ancilla qudit pair in the quantum systemcan be performed in parallel with the quantum measurement circuit.

At 608, the method can include determining reduced density matrix(s)from a set of the plurality of measurements. For instance, the classicalprocessor(s) 104 of FIG. 1 can determine reduce density matrix(s) from aset of the plurality of measurements. The set of plurality ofmeasurements can include a number of repeated measurements performed onthe quantum system. In some embodiments, the reduced density matrix(s)can include all k-qudit reduced density matrices for the quantum system.The number of repeated measurements can be determined as a function of asize of the subset of the quantum system described by the reduceddensity matrix irrespective of a number of quantum system qudits in thequantum system. The number of repeated measurements can be about(D+1)^(k)/∈², where D is the number of quantum levels associated witheach qudit, k is indicative of a number of qudits in the subset of thequantum system represented by the k-qudit reduced density matrix, and ∈is a desired precision of the k-qudit reduced density matrix.

At 610, the method can include outputting data indicative of the reduceddensity matrix(s). For example, the data indicative of the reduceddensity matrix(s) can be communicated to another component of thequantum computing system (e.g., another processor) or to a remotedevice. As another example, data indicative of the reduced densitymatrix(s) can be stored in one or more memory devices. As anotherexample, data indicative of the reduced density matrix(s) can beprovided for display on one or more display devices. As another example,data indicative of the reduced density matrix(s) can be provided foraccess by one or more computer programs, modules, executableinstructions, or the like.

FIG. 7 depicts a flow diagram of an example method 700 according toexample embodiments of the present disclosure. The method 700 can beimplemented using any suitable quantum computing system, such as thequantum computing system 100 depicted in FIG. 1 . FIG. 7 depicts stepsperformed in a particular order for purposes of illustration anddiscussion. Those of ordinary skill in the art, using the disclosuresprovided herein, will understand that various steps of any of themethods disclosed herein can be adapted, modified, performedsimultaneously, omitted, include steps not illustrated, rearranged,and/or expanded in various ways without deviating from the scope of thepresent disclosure.

At 702, the method can include accessing a quantum system (e.g., thequantum system 102 of FIG. 1 ). The quantum system can include one ormore quantum system qubits. The quantum system can include one or moreancilla qubits. Each ancilla qubit can be associated with one of the oneor more quantum system qubits.

At 704, the method can including configuring each ancilla qubit in aknown state. For instance, as discussed with reference to FIGS. 3 and 4, quantum gate(s) implemented by a quantum computing system canconfigure each ancilla qubit in the quantum system in a known state. Forthe case of qubits, each ancilla qubit can be configured in a knownstate μ (e.g., using one or more quantum gates), so that the totalsystem-ancilla state is ρ⊗μ^(⊗n).

In some embodiments, the ancilla qubit can be configured in a knownstate μ by rotating |0> about an x axis by an angle θ and a rotationaround the z axis by an angle ρ. If the Pauli-x, -y, and -z operatorsare sampled at the same rate, a natural choice of the ancilla state is

$\begin{matrix}{{\mu = {\frac{1}{2}\left( {{+ \frac{1}{\sqrt{3}}}\left( {\sigma^{x} + \sigma^{y} + \sigma^{z}} \right)} \right)}},} & (3)\end{matrix}$

which leads to Tr(σ^(x)μ)=Tr(σ^(y)μ)=Tr(σ^(z)μ)=1/√{square root over(3)}. In that regard, in some embodiments, θ is equal to about arccos(1/√{square root over (3)}) and ϕ is equal to about 3π/4.

At 706, the method can include obtaining a plurality of measurements(e.g., Bell-basis measurements) using a quantum measurement circuit. Theplurality of measurements can be performed, for instance, using thequantum circuit 320 depicted in FIG. 3 . The plurality of measurementscan be repeated a specific number of times. Measurements for each qubitand ancilla qubit pair in the quantum system can be performed inparallel with the quantum measurement circuit.

At 708, the method can include implementing a fermionic-to-qubitmapping. An example fermionic-to-qubit mapping is a ternary tree mappingas discussed above with reference to FIG. 4 . In the this mapping, thereare 3^(h)=2n+1 distinct root-to-leaf paths in the ternary tree, whereasthe total number of independent operators A_(p) is 2n. This is becausethe product of A_(p) for all paths p is proportional to the identityoperator. Therefore, 2n Majorana operators can be mapped to 2nindependent Pauli operators A_(p) with the same anticommutationrelations. The Pauli weight of A_(p) equals to the tree heighth=log₃(2n+1), which is lower than log₂ (n+1) in the BKT. As a result,the Pauli weights of elements in fermionic k-RDMs are at most2klog₃(2n+1).

At 710, the method can include determining reduced density matrix(s)from a set of the plurality of measurements. For instance, the classicalprocessor(s) 104 of FIG. 1 can determine reduce density matrix(s) from aset of the plurality of measurements. The set of plurality ofmeasurements can include a number of repeated measurements performed onthe quantum system In some embodiments, the reduced density matrix(s)can include all k-particle fermionic reduced density matrices for thefermionic quantum system. The number of repeated measurements can beabout (2n+1)k/∈², wherein n is a number of quantum system qubits, k isindicative of a number of particles in the subset of the quantum systemrepresented by the reduced density matrix, and E is a defined precision

At 712, the method can include outputting data indicative of the reduceddensity matrix(s). For example, the data indicative of the reduceddensity matrix(s) can be communicated to another component of thequantum computing system (e.g., another processor) or to a remotedevice. As another example, data indicative of the reduced densitymatrix(s) can be stored in one or more memory devices. As anotherexample, data indicative of the reduced density matrix(s) can beprovided for display on one or more display devices. As another example,data indicative of the reduced density matrix(s) can be provided foraccess by one or more computer programs, modules, executableinstructions, or the like.

Implementations of the digital and/or quantum subject matter and thedigital functional operations and quantum operations described in thisspecification can be implemented in digital electronic circuitry,suitable quantum circuitry or, more generally, quantum computationalsystems, in tangibly-implemented digital and/or quantum computersoftware or firmware, in digital and/or quantum computer hardware,including the structures disclosed in this specification and theirstructural equivalents, or in combinations of one or more of them. Theterm “quantum computing systems” may include, but is not limited to,quantum computers/computing systems, quantum information processingsystems, quantum cryptography systems, or quantum simulators.

Implementations of the digital and/or quantum subject matter describedin this specification can be implemented as one or more digital and/orquantum computer programs, i.e., one or more modules of digital and/orquantum computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The digital and/or quantumcomputer storage medium can be a machine-readable storage device, amachine-readable storage substrate, a random or serial access memorydevice, one or more qubits/qubit structures, or a combination of one ormore of them. Alternatively or in addition, the program instructions canbe encoded on an artificially-generated propagated signal that iscapable of encoding digital and/or quantum information (e.g., amachine-generated electrical, optical, or electromagnetic signal) thatis generated to encode digital and/or quantum information fortransmission to suitable receiver apparatus for execution by a dataprocessing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held, or stored in quantum systems, where thesmallest non-trivial system is a qubit, i.e., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states (e.g., qudits) are possible.

The term “data processing apparatus” refers to digital and/or quantumdata processing hardware and encompasses all kinds of apparatus,devices, and machines for processing digital and/or quantum data,including by way of example a programmable digital processor, aprogrammable quantum processor, a digital computer, a quantum computer,or multiple digital and quantum processors or computers, andcombinations thereof. The apparatus can also be, or further include,special purpose logic circuitry, e.g., an FPGA (field programmable gatearray), or an ASIC (application-specific integrated circuit), or aquantum simulator, i.e., a quantum data processing apparatus that isdesigned to simulate or produce information about a specific quantumsystem. In particular, a quantum simulator is a special purpose quantumcomputer that does not have the capability to perform universal quantumcomputation. The apparatus can optionally include, in addition tohardware, code that creates an execution environment for digital and/orquantum computer programs, e.g., code that constitutes processorfirmware, a protocol stack, a database management system, an operatingsystem, or a combination of one or more of them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL, Quipper,Cirq, etc..

A digital and/or quantum computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data, e.g., one or more scripts storedin a markup language document, in a single file dedicated to the programin question, or in multiple coordinated files, e.g., files that storeone or more modules, sub-programs, or portions of code. A digital and/orquantum computer program can be deployed to be executed on one digitalor one quantum computer or on multiple digital and/or quantum computersthat are located at one site or distributed across multiple sites andinterconnected by a digital and/or quantum data communication network. Aquantum data communication network is understood to be a network thatmay transmit quantum data using quantum systems, e.g. qubits. Generally,a digital data communication network cannot transmit quantum data,however a quantum data communication network may transmit both quantumdata and digital data.

The processes and logic flows described in this specification can beperformed by one or more programmable digital and/or quantum computers,operating with one or more digital and/or quantum processors, asappropriate, executing one or more digital and/or quantum computerprograms to perform functions by operating on input digital and quantumdata and generating output. The processes and logic flows can also beperformed by, and apparatus can also be implemented as, special purposelogic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or bya combination of special purpose logic circuitry or quantum simulatorsand one or more programmed digital and/or quantum computers.

For a system of one or more digital and/or quantum computers orprocessors to be “configured to” or “operable to” perform particularoperations or actions means that the system has installed on itsoftware, firmware, hardware, or a combination of them that in operationcause the system to perform the operations or actions. For one or moredigital and/or quantum computer programs to be configured to performparticular operations or actions means that the one or more programsinclude instructions that, when executed by digital and/or quantum dataprocessing apparatus, cause the apparatus to perform the operations oractions. A quantum computer may receive instructions from a digitalcomputer that, when executed by the quantum computing apparatus, causethe apparatus to perform the operations or actions.

Digital and/or quantum computers suitable for the execution of a digitaland/or quantum computer program can be based on general or specialpurpose digital and/or quantum microprocessors or both, or any otherkind of central digital and/or quantum processing unit. Generally, acentral digital and/or quantum processing unit will receive instructionsand digital and/or quantum data from a read-only memory, or a randomaccess memory, or quantum systems suitable for transmitting quantumdata, e.g. photons, or combinations thereof.

Some example elements of a digital and/or quantum computer are a centralprocessing unit for performing or executing instructions and one or morememory devices for storing instructions and digital and/or quantum data.The central processing unit and the memory can be supplemented by, orincorporated in, special purpose logic circuitry or quantum simulators.Generally, a digital and/or quantum computer will also include, or beoperatively coupled to receive digital and/or quantum data from ortransfer digital and/or quantum data to, or both, one or more massstorage devices for storing digital and/or quantum data, e.g., magnetic,magneto-optical disks, or optical disks, or quantum systems suitable forstoring quantum information. However, a digital and/or quantum computerneed not have such devices.

Digital and/or quantum computer-readable media suitable for storingdigital and/or quantum computer program instructions and digital and/orquantum data include all forms of non-volatile digital and/or quantummemory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; and CD-ROM and DVD-ROM disks; and quantumsystems, e.g., trapped atoms or electrons. It is understood that quantummemories are devices that can store quantum data for a long time withhigh fidelity and efficiency, e.g., light-matter interfaces where lightis used for transmission and matter for storing and preserving thequantum features of quantum data such as superposition or quantumcoherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a digital and/or quantumcomputer program product that includes instructions that are stored onone or more non-transitory machine-readable storage media, and that areexecutable on one or more digital and/or quantum processing devices. Thesystems described in this specification, or portions of them, can eachbe implemented as an apparatus, method, or electronic system that mayinclude one or more digital and/or quantum processing devices and memoryto store executable instructions to perform the operations described inthis specification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular implementations of the subject matter have been described.Other implementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results. As one example, theprocesses depicted in the accompanying figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In some cases, multitasking and parallel processingmay be advantageous.

1.-20. (canceled)
 21. A method, the method comprising implementing afermionic-to-qubit mapping using a ternary tree mapping.
 22. The methodof claim 21, wherein the method comprises determining a k-particlefermionic reduced density matrix based at least in part on the ternarytree mapping.
 23. The method of claim 22, wherein determining thek-particle fermionic reduced density matrix comprises performing anumber of repeated measurements on one or more qubits using a quantummeasurement circuit.
 24. The method of claim 23, wherein the number ofrepeated measurements is about (2n+1)^(k)/∈², where n is the totalnumber of quantum system qubits in the quantum system, k is indicativeof a number of fermionic modes in a subset of the quantum systemrepresented by the k-particle reduced density matrix, and ∈ is thedesired precision.
 25. The method of claim 21, wherein thefermionic-to-qubit mapping maps a single fermionic operator to a qubitoperator with a balanced number of Pauli-x, -y- and -z operators. 26.The method of claim 21, wherein the ternary tree comprises a pluralityof first nodes and a plurality of second nodes, wherein the second nodesare leaf nodes of the ternary tree.
 27. The method of claim 26, whereineach of the first nodes are associated with a qubit.
 28. The method ofclaim 21, wherein the second nodes are not associated with a qubit. 29.The method of claim 21, wherein there are 3^(h)=2n+1 distinctroot-to-leaf paths in the ternary tree, wherein h is the height of theternary tree, where n is a number of qubits in the fermionic-to-qubitmapping.
 30. The method of claim 21, wherein the Pauli weights ofelements in fermionic k-particle reduced density matrices arelogarithmic based on a number of qubits in the fermionic-to-qubitmapping.
 31. The method of claim 30, the Pauli weights of elements infermionic k-particle reduced density matrices are no more than2klog₃(2n−1), where n is a number of qubits in the fermionic-to-qubitmapping.
 32. A quantum computing system, comprising one or more qubits;and one or more processors operable to perform operations, theoperations comprising implementing a fermionic-to-qubit mapping using aternary tree mapping.
 33. The quantum computing system of claim 32,wherein the operations comprise determining a k-particle fermionicreduced density matrix based at least in part on the ternary treemapping.
 34. The quantum computing system of claim 33 whereindetermining the k-particle fermionic reduced density matrix comprisesperforming a number of repeated measurements on one or more qubits usinga quantum measurement circuit.
 35. The quantum computing system of claim34, wherein the number of repeated measurements is about (2n+1)^(k)/∈²,where n is the total number of quantum system qubits in the quantumsystem, k is indicative of a number of fermionic modes in a subset ofthe quantum system represented by the k-particle reduced density matrix,and c is the desired precision.
 36. The quantum computing system ofclaim 32, wherein the fermionic-to-qubit mapping maps a single fermionicoperator to a qubit operator with a balanced number of Pauli-x, -y- and-z operators.
 37. The quantum computing system of claim 32, wherein theternary tree comprises a plurality of first nodes and a plurality ofsecond nodes, wherein the second nodes are leaf nodes of the ternarytree.
 38. The quantum computing system of claim 32, wherein each of thefirst nodes are associated with a qubit.
 39. The quantum computingsystem of claim 32, wherein the second nodes are not associated with aqubit.
 40. The quantum computing system of claim 32, wherein there are3^(h)=2n+1 distinct root-to-leaf paths in the ternary tree, wherein h isthe height of the ternary tree, where n is a number of qubits in thefermionic-to-qubit mapping.
 41. The quantum computing system of claim32, wherein the Pauli weights of elements in fermionic k-particlereduced density matrices are logarithmic based on a number of qubits inthe fermionic-to-qubit mapping.